Closure of Subset of Closed Set of Topological Space is Subset/Proof 1

Theorem

Let $T$ = $\struct {S, \tau}$ be a topological space.

Let $F$ be a closed set of $T$.

Let $H \subseteq F$ be a subset of $F$.

Let $H^-$ denote the closure of $H$.

Then $H^- \subseteq F$.

Proof

\(\ds H\)

\(\subseteq\)

\(\ds F\)

\(\ds \leadsto \ \ \)

\(\ds H^-\)

\(\subseteq\)

\(\ds F^-\)

Topological Closure of Subset is Subset of Topological Closure

\(\ds \)

\(=\)

\(\ds F\)

Set is Closed iff Equals Topological Closure

$\blacksquare$